11 problems

Let $\rho$ solve the advection--diffusion equation on the two-dimensional torus $\mathbb{T}^2=[-\pi,\pi]^2$, with divergence-free velocity field $\mathbf{u}$, diffusivity $\mu$, an…

Strictness characterization.

Steady-state convergence conjecture. The solution $\mathbf{u}$ converges towards a steady state.

Strong conjecture. The heat transfer satisfies

Weak conjecture. The heat transfer satisfies

Gradient blow-up conjecture. Failure to include non-locality in the advection term, equivalently setting $K$ to be a Dirac delta function, will lead to gradient blow-up.

Let $P^\varepsilon$ and $\overline P^\varepsilon$ be the time-$1$ solution operators of the Lagrangian and averaged advection-diffusion equations, respectively. Write … The operato…

Let $M$ be the material manifold, let $u_\varepsilon$ solve the Lagrangian advection-diffusion equation, and let $\overline u_\varepsilon$ solve the averaged equation … where…

Fix $p\in[1,+\infty)$. Let $b\in C^{\infty}([0,T]\times\mathbb{T}^d)$ be divergence free and let $u_0\in W^{1,2}(\mathbb{T}^d)$. If $u^\nu$ is a solution of the advection diffusion…

Let $d\geq 2$ and $p>2$ be fixed. For a divergence free velocity field $b\in L^{\infty}([0,1],W^{1,p}(\mathbb{T}^d,\mathbb{R}^d))$ and an initial datum…

Cantrell-Cosner-Lou concentration conjecture. $u$ concentrates precisely on the set of positive local maximum points of $m(x)$ as $\alpha\rightarrow\infty$.